A Linear Extension of the Martingale Convergence Theorem

Abstract

Let $X_1, X_2, \cdots$ be a sequence of random variables satisfying $E(X_{n + 1}\mid X_n, X_{n - 1}, \cdots, X_1) = a_1 X_n + a_2 X_{n - 1} + \cdots + X_{n - k - 1}, n \geqq k$, where $a_1 + a_2 + \cdots + a_k = 1$. Under certain general conditions, mainly that $\sup_nE|X_n| < \infty$, it is shown that $X_n - Y_n \rightarrow\operatorname{a.s.} 0$, where $\{Y_n\}$ is a solution of the homogeneous equation $y_n = a_1y_{n - 1} + a_2y_{n - 2} + \cdots + a_ky_{n - k}$. Several applications of possible theoretical interest are described. Also, the results suggest some extensions of classical results in the theory of random walks which are outlined.